Remark 3.2 Corollary REF  was first shown by Xu in [1]}. There he constructed a highly lacunary Fourier series to show that all Walsh-Paley martingales satisfied (REF ) only by using the Poisson semigroup on the unit disk. Hence, from his proof, \(X\)  is martingale cotype \(q\)  iff the Poisson semigroup satisfies (REF ).

